Tuesday, February 16, 2016

Day 101: Slope Dude, Scatterplot Predictions & Factoring Shortcuts

Today's estimation was how many sweethearts are in a package. I liked when students saw that it was 28 grams, so they estimated that each candy weighed a gram, so 28 grams was their estimate. It was definitely close. I like that students knew there would be some empty space so they anticipated it not being as full as they thought it would be.




We watched the video "Slope Dude Says." Watch it above. Afterwards, I asked students why a horizontal line has a slope of zero. They said it's because there's no vertical change. Then I asked how a slope could be undefined? This was harder for students to articulate, and some students saw that there was zero horizontal change, and it rises, so it's any number over zero, but you can't divide by 0, so it's undefined.
Relating slope to the comparison between rise and run in a graph.
Then we played Slope Dude Says, as recommended by Sarah at the blog Math Equals Love. I should have asked students like I did in 5th period first, how could you show zero slope with your hands. They put their hands out like a horizontal line. Then I asked for positive and they put their right hand up and left hand down. They did the opposite for negative. They realized undefined would be hands at your side. Then we played where I called it out, and they sat down if they flinched. Just a math version of Simon Says.

Then students worked on a CPM problem called Personal Trainer. They analyze a table of values with minutes and miles cycled. Students realized it wasn't growing at a constant rate, but I had to remind students to give evidence from the table of values to back up their statement. I liked when students said if you took 5 minutes to bike 1 mile, you should take 10 minutes to bike 2 miles, but the table has 8 minutes for 2 miles. Students knew when one quantity doubles, the other must double if it is proportional. Some also looked at the unit rates.

Then they graphed their scatterplot with miles being the dependent variable. When they drew their trend line, I asked them if it had to be in a certain place to make sense. A few students reasoned it had to go through the origin, because in 0 time, you'd cycle 0 miles. Then they predicted how long it would take to cycle 10 miles. I asked students to show it on their graph with a dashed line, as per Ms. Demailly's suggestion she made to her students.

Then they drew a slope triangle through the lattice points and interpreted it's speed. Some students ignored how they scaled their axes when labeling their slope triangles. Here you can see Ethan got a slope of 2/6 which means 2 miles cycled in 6 minutes.


Here, Megan's slope was 2 miles cycled in 7.5 minutes, or 1 mile in 3.75 minutes. Some went down to what fraction of a mile was cycled in 1 minute.
In accelerated, each group got 1 or 2 paper strips to write a quadratic function, and factor it on the other side. Then they taped it on the board. We then looked at them as a class and grouped them into categories. The first group had the same factors, except one was added and one was subtracted. We later labeled this difference of squares. The second group was same factors squared. The book introduced this as perfect square trinomials. The third category had factors that weren't the same, and the final category was x^2+4 which was not factorable. Some students wondered why it wasn't factorable. So, I told them to imagine its graph and where the graph doesn't go through.

The categories.

Then students worked on the practice problems. As I went around, there wasn't a lot of collaboration and discussion so some students were admittedly lost and didn't understand the shortcuts. So, we came together as a class and looked at 4x^2-25. They saw that both are square numbers, so the first factors are (2x   ) and (2x    ). Then they thought -5 and 5 because that multiplies to -25. I asked them why there was no middle term. They said it cancelled out.

Then we looked at perfect square trinomials. Basically, 4x^2+20x+25 is similar, (2x+5)(2x+5), except the middle terms don't cancel, they add together to get 20x.

Here we didn't factor it all the way, but we reviewed putting the first and last terms as one diagonal. Then they constructed the top and bottom of the diamond, the top being the product of the diagonal and the bottom being the middle term of the trinomial. When factoring the sides of the generic rectangle you get (7x+1)(x-3)

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